In a genteel Soho dining room, with morning tea-drinkers clinking and murmuring all around us, I ask Tilda Swinton if she has ever felt monstrous. She tilts her wholesome head-boy face, with its short flop of blond hair. "Does one ever feel it," she says, "or is one simply monstrous? I mean, I've been monstrous, I think." Until recently, she says, she had a reputation in her family for having saved her younger brother's life as a child, when she had actually intended to kill him.

"I was going to kill him because he was a boy, naturally," she says, "and I already had two brothers, and that was just too much to bear." She was four and a half when she entered his room, morbidly determined. "I hadn't thought it through, but I was willing to wing it. And I noticed he had a ribbon from a baby bonnet sticking from the corner of his mouth. I started to pull it out – and then was witnessed in this great act of love, of nurture!"

At a nearby table a baby starts screaming, its wails vibrating through the Earl Grey teapots. Monstrousness, Swinton realised, might always be bubbling just beneath a person's surface. "Whenever there are children killing children, or perpetrating great acts of violence, there's always this word 'evil' pulled out of the top drawer. Not even the bottom drawer. It's a very quick response. And I'm always struck by it, because from the age of four and a half I have known that it ain't in no drawer. It's at closer hand. Isn't that the triumph of civilisation? That we manage not to be monstrous?"

At 50, Swinton's face still shifts effortlessly between ages, sexes, aesthetics. One minute it bears traces of her great-grandmother, Mrs George Swinton, a society beauty painted by John Singer Sargent in the late 1800s; the next, a bare skull. Either way, it's hard not to stare. Back in 1995, she exhibited herself lying in a glass case at the Serpentine gallery in London, which now seems slightly ironic since, of all the women actors in the world, she must be the least easy to objectify, to box in. She is cool, intelligent, engaging with questions deeply and happily when she likes them, and chewing them up roughly when she doesn't. She chides imprecision, asks for a rephrase, has a frightening talent to make you feel small.

We meet hours after a screening of Swinton's new film, We Need To Talk About Kevin. Directed by Lynne Ramsay, it is beautifully shot yet utterly grim, like a shovel to the back of the head. Swinton is Eva Khatchadourian, a successful travel writer presiding over her own publishing company, her world privileged and expansive, who decides to have a child in her late 30s, despite misgivings over how it might affect her blissful life with husband Franklin. She gives birth to Kevin, the bond between mother and son fails to materialise, and he becomes the breathing, spitting embodiment of her resentment at how her life has changed and narrowed – until, at 15, he commits a violent atrocity.

Swinton is excellent. Appearing in almost every frame, her face is im-passive, yet somehow it's explicit that a twister is roiling beneath her skin. She describes the film as a Greek tragedy, "the play Euripides didn't have the balls to write", and says she became interested in the part when the script began tending towards "dumbness … I'm not really particularly good at being articulate. I'm much more wired for the dumb look." She has said she was pleased when someone once compared her to Buster Keaton.

The novel the film is based on, by Lionel Shriver, is distinctly wordy, a collection of letters from Eva to her husband after the carnage, words spilling out like hounds, panting after answers. The film, instead, is quiet, concentrating on the space between characters, all the bitterness and suspicions that remain unsaid. The idea of Eva writing endless letters on screen, or describing her life in voiceover, was quickly rejected, says Swinton, "because there's nobody to talk to. It's indescribable. It's unspeakable, what she's going through. It's not even loneliness. It's purgatory, which is much worse."

There was a moment, she says, "when we talked about shooting away from Kevin, so you never saw him, in a similar way to Jaws". This would have been more in keeping with the book, in which Kevin is always kept at one small remove, by being described through his mother's eyes. His intense, unrelenting nastiness makes more sense in the novel, because of her obvious retrospection: given the nature of Kevin's crime, it's only natural that she focuses forensically on their every spiteful meeting.

But the film hops back and forth between carnage and babyhood, assorted scenes from a life, and there is a baldness to Kevin's evil that makes the character seem grotesque. As a child he defecates at will, just to rile his mother; he masturbates mechanically with the bathroom door open, sneering at Eva over his shoulder.

Swinton says she doesn't feel the word evil is right: "It's a disclaimer, that word, even the concept of it." Instead, she sees the film as a fantasy, in which "he's playing out her detachment. His lack of empathy is her detachment. All those early incidents she feels so tortured by, like mocking him through the bars of his – cage, I was going to say – but the bars of his cot, saying, 'Mummy was happy before little Kevin came along, now Mummy wakes up every morning and wishes she was in France' … That early image of her holding up the baby, when he's screaming, and that terrible, inauthentic smile she gives.

"I mean, wouldn't you scream if you had a mother doing that? I would! It's that feeling of inauthenticity, of her having edited out the majority of herself in this relationship; and it's hard not to sympathise – to use that word – with his, um, efforts to get her attention."

'OK, fine, now I'm a girl'

Does Swinton think the film might put women off having children, as has been suggested of the book? "I don't see why it would," she says sunnily. "It's a fantasy. It's never going to be that bad … Everybody thinks for one moment when they're pregnant that they're actually carrying the spawn of the devil." Did she? Swinton's twins, Xavier and Honor, are now in their teens. "Oh yeah, of course, because you're in freefall, so you have to wonder. It would be crazy not to."

She continues: "But I remember noticing, when I had my babies, how much I liked them, and not just loved them, but I was really into them. I knew I was going to be curious about them, and up for the mayhem ahead. But at the same time I remember noticing I was relieved this thing was present in me. And I hadn't realised there might be a doubt. I thought it was going to be automatic – and something in me said, 'No, you're really lucky here.'"

Swinton's life to date has fallen into two distinct halves: the first at the heart of the establishment, the second well outside it. She was born to Major-General Sir John Swinton, into the landed classes, and was sent to boarding school at 10. This is the age, she thinks, when children are just working out what makes them distinctive, where they fit into their families. "So it's a weird moment to go, 'OK, we're going to distract you from all that useful evolutionary work, and lump you together with a lot of people on a desert island called school and leave you to it.'" She was excited at first, "because I was happy to be amongst girls, having been in a family with a bunch of boys. I was going, 'OK, fine. So now I'm a girl.'" It's reminiscent of her breakthrough role as Orlando, in 1992, a character who goes to bed a man, wakes up a woman, and says: "Same person. No difference at all."

But overall, she really didn't enjoy boarding school. She was bullied and homesick: "I don't think I spoke for five years." I've read that the girls were expected to become wives of the establishment, a notion borne out by the marriage of her classmate, Lady Diana Spencer. I ask if there was a sense of being groomed for marriage, and she bristles. "I'm wondering about the word 'groomed'. I'm also wondering that we're sitting talking, for the Guardian, about Diana Spencer. I can't quite believe it." We bat this about a bit. I explain I'm interested in the wider idea of women being expected simply to marry. "It was a holding bay," she says finally. "We managed to survive. Most of us."

She seems similarly tetchy when I ask about her relationship with radical director Derek Jarman, with whom she made seven films before his death in 1994. They first met in 1986, when he was casting for Caravaggio, and she was at a turning point, in her mid-20s. She had studied English at Cambridge, gone on to the RSC, and was on the verge of giving up performing. "I was looking for film, always. I only worked in the theatre because my friends were working there. I just slid sideways into it, and at a certain point realised it was absolutely not what I was interested in."

Comfort with mayhem

I have never read a description of her first meeting with Jarman, and I'm interested in how it played out, this decisive step away from her establishment path. But when I ask, she grimaces. "You're doing a really funny thing," she says tightly, "which is that you're quoting back to me things that I remember I've said before, so I feel like saying, 'Yes, tick, tick, tick.'" She sighs. "Fine. Right, I met Derek and I met practising artists for the first time, and he was the first film-maker. I think he was almost the first living artist I'd ever met."

In the years since, she has moved through art films, big-budget films, to a standing where she can do just about anything. In 2008, she won an Oscar for her performance in Michael Clayton, as a corrupt company's chief legal counsel, desperately mimicking calm in the face of meltdown. Again, Swinton's face was impassive, while her character jangled behind it like a bead curtain.

The key to Swinton's career, her parenting, her life – the phrase she returns to repeatedly – is comfort with mayhem, wildness beneath the surface. Kevin was made in 30 days, a punishing pace that would upset many actors. Not Swinton. "If you've got an enormous budget, and you've written a scene for bright sunlight, and it starts to rain, you're going to wait three weeks for it to stop. But if you haven't, you're going to have to make friends with chaos. That's the main difference. And that's something I know in my marrow."

We finish our tea, the baby keeps wailing, and Swinton strides off – as boyish, girlish, extraordinary as ever.

 This article was amended on 12 and 13 October 2011. The original stated that John Singer Sargent painted Swinton's great grandmother at the end of the 1900s. The original also said Swinton played "a businesswoman running a corrupt company" in the film Michael Clayton. This has been corrected.